Resurgence out of the (literal) box Aleksey Cherman INT, University of Washington work in progress with M. Unsal and D. Dorigoni
Resurgence for QFT Belief: QFT observables are transseries in the couplings Generically, all series are separately divergent and ambiguous, but 𝒫 ( λ ) is well-defined due to devious conspiracies between terms Why believe this specifically in full QFT? Very h ard to explore high loop orders!
Resurgence for 0d QFT First explicit check: 0-dimensional “QFT” Can be done very explicitly. Resurgence idea works!
Resurgence for QM Second explicit check: 1-dimensional “QFT” - quantum mechanics! Detailed explorations focused on QM with smooth potentials V(x) Dunne perturbation theory + finite # of conditions on 𝝎 (x) = everything. + Unsal 2013: Resurgence idea works!
Resurgence for QM Second explicit check: 1-dimensional “QFT” - quantum mechanics! Detailed explorations focused on QM with smooth potentials V(x) Resurgence idea works! Dunne perturbation theory + finite # of conditions on 𝝎 (x) = everything. + Unsal 2013: Relation of resurgence to Basar + Dunne 2015 elliptic curve associated to V(x) Gives some explanation of `why’ it works; similar story can be told in 0d.
Witten 2009; Dunne, Unsal, Resurgence for QFT? AC, Dorigoni, Basar, … 2013-now Why should the d = 1 results generalize to d > 1? Elliptic curve picture seems closely tied to QM, generalization unclear. Path integral perspective? “Lefshetz thimble” integration cycles One `thimble’ per critical point of classical action, defined by steepest descent. Z( λ ) = + - + +… perturbation theory non-perturbative contributions {set of thimbles} = complete basis for convergent path integrals Resurgence relations = jumps in C k as arg[ λ ] varies.
Resurgence for QFT? Thimble perspective might sound taylor-made for generalization to QFT… … but this isn’t obvious! Witten proved thimble decomposition works in d = 1 > 0 No proof that set of critical-point cycles is a basis away from d = 1! Several possibly-related issues. What counts as a critical point? How to perform decomposition? … Behtash, Dunne, Even in d = 1 discontinuous saddle-point-field Schafer, Sulejmanpasic, configurations must be taken into account! Unsal, 2015 Construction in d > 1 may be sensitive to regularization of integral. Shouldn’t be too shocking: regularization always important in d > 1 !
Resurgence in QFT Third explicit check: 1+1D asymptotically-free QFTs CP N-1 , principal chiral, O(N), and Grassmannian non-linear sigma models To the extent it’s been checked, resurgence works! linear combinations of Dunne, Unsal, AC, Dorigoni 2012-2015 Why the weasel words? In d > 1 QFT, very difficult to precisely characterize large-order behavior Strong coupling in IR in asymptotically-free theories All work so far used idea of adiabatic compactification from R 2 to RxS 1
Tiny boxes as tools Compactify asymptotically-free QFT from R D to R D-1 xS 1 Idea: when S 1 size L << Λ -1 , theory becomes ≈ weakly-coupled R D-1 S 1 Simplest circle is a thermal one. Trouble: physics at small-L and large-L can look totally different Examples: Large N phase transitions as a function of L Dependence of gap Δ on 2D strong scale Λ is power law at large L, only logarithmic at small L.
Adiabatic small circle limit For a smooth L << Λ -1 limit, use special non-thermal boundary conditions. Idea is actually quite general, very closely related to constructions in 4D gauge theory Unsal and collaborators, 2012-onward 4D gauge theory: adiabatic small-L limit obtained with Z N - invariant S 1 holonomy for the dynamical gauge field 2D sigma models: adiabatic small-L limit obtained with Z N - invariant S 1 holonomy for the background `flavor’ gauge field With such compactifications, effective KK scale is 1/(NL), not 1/L. Large N and small L limits do not commute - tied to large N volume independence!
Coupling flow with adiabatic compactification λ large N 1 volume independence Flow for NL Λ ≪ 1 Flow for NL Λ ≫ 1 Semiclassically calculable regime λ ( 1 / NL ) Q ( N L ) - 1 Λ NL Λ >> 1 regime is strongly coupled The NL Λ << 1 regime gives a weakly-coupled theory Physics is very rich - mass gap, renormalons present at small N L!
Resurgence in a box In perturbation theory 2D sigma models like O(N), CP N-1 , etc are gapless. What about non-perturbatively, in the small NL Λ limit? Need to know about non-perturbative saddle points! The Z N -invariant holonomies make instantons fractionalize into ~ N constituent `fractons’ (or `monopole-instantons’, etc.) Dabrowski, Without instantons, what fractionalizes are `unitons’ - Dunne; AC, Dorigoni, finite-action, non-BPS saddle-point solutions. Dunne, Unsal Very common in 2D: relevant homotopy group is π 2 . O(N) model: π 2 [O(N)] = 0; SU(N) Principal chiral model π 2 [SU(N)] = 0 The fractons, or composites built from them, drive appearance of mass gap!
Fractionalization of unitons Uniton action Fracton action density density SU(2) SU(3)
Resurgence in a box To obtain results, use small NL Λ 1D effective field theory. EFT UV cutoff μ ～ 1/(NL). At small NL Λ , mass gap ends up looking like Fluctuations Schematic expression: really there’s log( λ ) factors, and sometimes gap starts at with contributions from two fractons, etc The series appearing above are resurgent.
AC, Dorigoni, Resurgence in a box Unsal coming soon So, seems resurgence applies to 2D QFTs — at least to leading order. But the check used that small-L EFT, which is QM. From the perspective of earlier worries, this is a bit of a cheat! A demonstration directly in d = 2, without compactification, would be better.
AC, Dorigoni, Resurgence in full QFT Unsal coming soon Warning: work in progress from here onward! Use large N expansion to get around strong-coupling issues on R 2 Idea is to work perturbatively in 1/N, but exactly in ’t Hooft coupling, then explore ’t Hooft coupling expansion structure. Example for this talk: 2D O(N) model Results generalize to other vector-like NLSMs
Resurgence in large N O(N) model Integrate in a Lagrange multiplier σ to make life easier: Questions: what’s the mass gap Δ ? Resurgence as a function of λ ? Perturbation theory: theory of N - 1 massless particles, Δ = 0. To define theory, must regularize UV. We’ll use momentum cutoff μ . Mass gap physics far outside any semiclassical regime on R 2 !
Resurgence in large N O(N) model Large N solution is textbook material - see e.g. Peskin & Schoeder Integrate out n a fields, giving At large N, physics captured by saddle-point for σ , which satisfies Want σ in terms of μ and λ . Non-zero σ is a mass-squared for n a fields!
Resurgence in large N O(N) model The textbooks all say that Spectrum has N massive particles, with m 2 = σ Celebrated result: O(N) beta function is one-loop exact at large N
Resurgence in large N O(N) model Compare large N result on R 2 to adiabatic-small-L expectation: versus Fluctuations Large N limit suppresses fluctuations and kills multi-fractons!? Conceivable… But is it true?
AC, Dorigoni, Resurgence in large N O(N) model Unsal coming soon The textbooks all say that Bizarre fact: the equal sign is wrong. Consequences: non-perturbative corrections!
AC, Dorigoni, Coupling constant flow Unsal coming soon One-loop coupling diverges at μ = Λ = e -1/2 λ : Exact large N coupling only diverges at μ = 0:
AC, Dorigoni, Coupling constant flow Unsal coming soon Coupling 3.0 one-loop λ 2.5 large N λ 2.0 1.5 1.0 0.5 10 μ 0 2 4 6 8
AC, Dorigoni, Resurgence in large N O(N) model Unsal coming soon Compare large N result on R 2 to adiabatic-small-L expectation: R 2 versus Small L, R x S 1 , N < ∞ Fluctuations Large N limit still suppresses fluctuations; but way closer resemblance! Are the `fractons’ somehow surviving all the way to strong coupling?
AC, Dorigoni, Exact large N mass gap & coupling Unsal coming soon We’re still confused on what to make of all this. Well known that only first two coefficients of beta functions invariant under scheme changes. More precisely, first two coefficients of series expansion of beta function invariant under scheme changes represented by power series. Still trying to understand whether any extra `non-perturbative universality’ can be revealed by trans-series perspective. In any case, tantalizing that exact large N result has some interesting properties + resonance with small-L studies.
AC, Dorigoni, Unsal O(N) model at large N coming soon; also F. David 1984 So far, we have a transseries but no resurgence, due to suppression of fluctuations by large N To see resurgent behavior, need to look at 1/N corrections. To be specific, we’ll continue to examine < σ >
AC, Dorigoni, Unsal O(N) model at order 1/N coming soon; also F. David 1984 Large N theory consists of N massive fields with mass m = Δ a b and a field ` σ ’ describing fluctuations around VEV, σ → < σ > + σ /N 1/2 with an interaction vertex a Dependence on λ only enters through m! b
AC, Dorigoni, Unsal O(N) model at order 1/N coming soon; also F. David 1984 Leading correction to < σ > comes from x The 1/N correction is UV-divergent. Put cutoff at μ , assume μ ~ N 0
Recommend
More recommend
